Finite Group Databasebeta

← #27287509

Subgroups — #27287509

Order 13568.

520 conjugacy classes of subgroups.

The subgroup lattice has 440 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20279710C211yesmaximal, characteristic
2#20275676C211yesmaximal, characteristic, complemented
3#20275473C211yesmaximal, characteristic, complemented
4#20274716C211yesmaximal, characteristic
5C2×#16810370C211yesmaximal, characteristic, complemented
6C2×#16810352C211yesmaximal, characteristic
7C53×#1279C211yesmaximal, characteristic, complemented
8C2×#13133217C2211yescharacteristic
9C2×#13133205—21no—
10C2×C4×D212C2211yescharacteristic
11C2×#13133150—21no—
12C2×C4⋊#88322C2211yescharacteristic
13C2×#13133139—21no—
14#16810550—22no—
15#16810529—21no—
16#16810493C2211yescharacteristic
17#16810480—21no—
18#16810480—21no—
19#16810386—21no—
20#16810386—22no—
21#16810385—21no—
22#16810370—22no—
23#16810352—22no—
24C2×C4⋊C8×C53C2211yescharacteristic
25C2×C4×C8×C53C2211yescharacteristic
26C4×C4⋊C4×C53C2211yescharacteristic
27C22×Dic106—21no—
28C22×D212D411yescharacteristic
29C22×C4×D53—21no—
30C22×#88322—21no—
31#13133217—21no—
32#13133217—21no—
33#13133217—21no—
34#13133217—21no—
35#13133205—42no—
36#13133205—42no—
37C4×D212—21no—
38C4×D212—21no—
39C4×D212—21no—
40C4×D212—21no—
41C2×C4⋊Dic53—21no—
42C2×C4⋊Dic53—21no—
43C2×C4⋊Dic53—21no—
44C2×C4⋊Dic53—21no—
45C2×C4⋊Dic53D411yescharacteristic
46C2×C4⋊Dic53D412yes—
47C2×#11719921—22no—
48C2×#11719921—22no—
49C2×#11719921—21no—
50#13133150—42no—
51C4⋊#88322—21no—
52C4⋊#88322D412yes—
53#13133139—42no—
54#13133124—22no—
55#13133124—22no—
56C22×C8×C53—21no—
57C22×C8×C53—21no—
58C2×C4⋊C4×C53D411yescharacteristic
59C2×C4⋊C4×C53D411yescharacteristic
60C2×C42×C53—21no—
61C2×C42×C53C2311yescharacteristic
62C4⋊C8×C53—21no—
63C4⋊C8×C53—21no—
64C4×C8×C53—21no—
65C4×C8×C53—21no—
66#95×C53—21no—
67C23×D53—21no—
68C22×C4×C53—21no—
69C22×C4×C53—21no—
70C22×C4×C53C2×D411yescharacteristic
71C22×C4×C53C2×D411yescharacteristic
72C22×C4×C53C2×D411yescharacteristic
73C22×Dic53—21no—
74C22×Dic53—22no—
75C22×Dic53—41no—
76C2×D212—21no—
77C2×D212—21no—
78C2×D212—21no—
79C2×D212—21no—
80C2×D212—21no—
81C2×D212D812yes—
82C2×C4×D53—41no—
83C2×C4×D53—41no—
84C2×C4×D53—41no—
85C2×C4×D53—41no—
86C2×C4×D53—41no—
87C2×C4×D53—41no—
88C2×Dic106—41no—
89C2×Dic106—41no—
90C2×Dic106—41no—
91C2×Dic106—41no—
92C2×Dic106—41no—
93C2×Dic106—41no—
94C2×C8×C53—22no—
95C2×C8×C53—41no—
96C2×C8×C53—41no—
97C2×C8×C53—41no—
98C2×C8×C53—41no—
99C2×C8×C53—41no—
100C4⋊C4×C53—21no—
101C4⋊C4×C53—21no—
102C4⋊C4×C53—21no—
103C4⋊C4×C53—21no—
104C42×C53—41no—
105C42×C53—41no—
106C42×C53C2×D411yescharacteristic
107C42×C53#4111yescharacteristic
108C42×C53#4111yescharacteristic
109C42×C53#4111yescharacteristic
110#11719921—44no—
111#11719921—44no—
112#11719921—41no—
113#11719921—41no—
114C4⋊Dic53—21no—
115C4⋊Dic53—22no—
116C4⋊Dic53—21no—
117C4⋊Dic53—22no—
118C4⋊Dic53—41no—
119C4⋊Dic53—41no—
120C4⋊Dic53—42no—
121C4⋊Dic53—41no—
122C4⋊Dic53—41no—
123C4⋊Dic53—41no—
124C4⋊Dic53—41no—
125C2×#88322—22no—
126C2×#88322—41no—
127C2×#88322—41no—
128C23×C53#12011yescharacteristic
129C22×D53—22no—
130C22×D53—41no—
131C22×D53—41no—
132C22×D53—41no—
133C22×D53—41no—
134C22×D53—41no—
135C22×D53—41no—
136C2×C4×C53—21no—
137C2×C4×C53—21no—
138C2×C4×C53—41no—
139C2×C4×C53—41no—
140C2×C4×C53—41no—
141C2×C4×C53—41no—
142C2×C4×C53—41no—
143C2×C4×C53—41no—
144C2×C4×C53#12111yescharacteristic
145C2×C4×C53#12111yescharacteristic
146C2×C4×C53#12111yescharacteristic
147C2×C4×C53#12311yescharacteristic
148C2×C4×C53#12311yescharacteristic
149C2×C4×C53#12411yescharacteristic
150C2×C4×C53C2×D811yescharacteristic
151C2×C4×C53#13511yescharacteristic
152C2×C4×C53#13511yescharacteristic
153C2×C4×C53#13511yescharacteristic
154C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
155C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
156C2×C4×C53#13711yescharacteristic
157C2×C4×C53#13711yescharacteristic
158C2×Dic53—42no—
159C2×Dic53—41no—
160C2×Dic53—42no—
161C2×Dic53—41no—
162C2×Dic53—42no—
163C2×Dic53—41no—
164C2×Dic53—81no—
165C2×Dic53—81no—
166C2×Dic53—81no—
167D212—22no—
168D212—22no—
169D212—41no—
170D212—41no—
171C4×D53—81no—
172C4×D53—81no—
173C4×D53—81no—
174C4×D53—81no—
175Dic106—81no—
176Dic106—81no—
177Dic106—81no—
178Dic106—81no—
179C8×C53—42no—
180C8×C53—81no—
181#88322—42no—
182#10416—531nomaximal, Sylow
183C22×C53#39411yescharacteristic
184C22×C53#44711yescharacteristic
185C22×C53#44911yescharacteristic
186C22×C53#45211yescharacteristic
187C22×C53#45211yescharacteristic
188C22×C53#45411yescharacteristic
189C22×C53#45611yescharacteristic
190C2×D53—42no—
191C2×D53—42no—
192C2×D53—42no—
193C2×D53—81no—
194C2×D53—81no—
195C2×D53—81no—
196C2×D53—81no—
197C4×C53—21no—
198C4×C53—21no—
199C4×C53—81no—
200C4×C53—81no—
201C4×C53#46311yescharacteristic
202C4×C53#46411yescharacteristic
203C4×C53#46611yescharacteristic
204C4×C53#46811yescharacteristic
205C4×C53#48011yescharacteristic
206C4×C53#48211yescharacteristic
207C4×C53#48811yescharacteristic
208C4×C53#48911yescharacteristic
209Dic53—82no—
210Dic53—81no—
211Dic53—161no—
212#2547—531no—
213#1990—531no—
214#1947—531no—
215#1802—531no—
216C2×#332—531no—
217C2×D4⋊C8—531no—
218#1279D5311yescharacteristic, complemented
219C2×C53#162511yescharacteristic
220C2×C53#163811yescharacteristic
221C2×C53#164711yescharacteristic
222C2×C53#165211yescharacteristic
223C2×C53#203311yescharacteristic
224C2×C53#203811yescharacteristic
225C2×C53#204211yescharacteristic
226D53—82no—
227C2×C4⋊Q8—531no—
228C2×#124—1061no—
229C2×C4×D4—531no—
230C2×#107—1061no—
231C2×C4⋊C8—531no—
232C2×C4⋊C8C2×D5311yescharacteristic
233C2×#102—1061no—
234C2×C4×C8C2×D5311yescharacteristic
235#397—1062no—
236#391—1061no—
237#384—531no—
238#380—1061no—
239#380—1061no—
240C4×C4⋊C4C2×D5311yescharacteristic
241#340—1061no—
242#340—1062no—
243#340—1061no—
244#332—1062no—
245D4⋊C8—1062no—
246C53#1041611yescharacteristic, complemented, Sylow
247C22×Q8—1061no—
248C22×D4—531no—
249C23×C4—1061no—
250C22×C8—21no—
251C22×C8—21no—
252C22×C8—1061no—
253C4⋊Q8—1061no—
254C4⋊Q8—1061no—
255C4⋊Q8—1061no—
256C4⋊Q8—1061no—
257#124—2122no—
258#124—2122no—
259C4×D4—1061no—
260C4×D4—1061no—
261C4×D4—1061no—
262C4×D4—1061no—
263C2×C4⋊C4—531no—
264C2×C4⋊C4—532no—
265C2×C4⋊C4—1061no—
266C2×C4⋊C4—1061no—
267C2×C4⋊C4—1061no—
268C2×C4⋊C4—1061no—
269C2×C4⋊C4D21211yescharacteristic
270C2×C4⋊C4D21211yescharacteristic
271C2×C22⋊C4—1062no—
272C2×C22⋊C4—1062no—
273C2×C22⋊C4—1061no—
274C2×C42—21no—
275C2×C42C22×D5311yescharacteristic
276#107—2122no—
277C4⋊C8—21no—
278C4⋊C8—21no—
279C4⋊C8—532no—
280C4⋊C8—1061no—
281#102—2122no—
282C4×C8—21no—
283C4×C8—21no—
284#95—21no—
285#95—1062no—
286#95—1062no—
287C24—1061no—
288C2×Q8—2121no—
289C2×Q8—2121no—
290C2×Q8—2121no—
291C2×Q8—2121no—
292C2×Q8—2121no—
293C2×Q8—2121no—
294C2×D4—532no—
295C2×D4—1061no—
296C2×D4—1061no—
297C2×D4—1061no—
298C2×D4—1061no—
299C2×D4—1061no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—1061no—
303C22×C4—1062no—
304C22×C4—2121no—
305C22×C4—2121no—
306C22×C4—2121no—
307C22×C4—2121no—
308C22×C4—2121no—
309C22×C4—2121no—
310C22×C4—2121no—
311C22×C4C2×D21211yescharacteristic
312C22×C4D4×D5311yescharacteristic
313C22×C4D4×D5311yescharacteristic
314C2×C8—22no—
315C2×C8—41no—
316C2×C8—41no—
317C2×C8—41no—
318C2×C8—41no—
319C2×C8—41no—
320C2×C8—1062no—
321C2×C8—2121no—
322C2×C8—2121no—
323C4⋊C4—21no—
324C4⋊C4—21no—
325C4⋊C4—21no—
326C4⋊C4—21no—
327C4⋊C4—1061no—
328C4⋊C4—1062no—
329C4⋊C4—1061no—
330C4⋊C4—1062no—
331C4⋊C4—2122no—
332C4⋊C4—2121no—
333C4⋊C4—2121no—
334C4⋊C4—2121no—
335C4⋊C4—2121no—
336C4⋊C4—2121no—
337C4⋊C4—2121no—
338C22⋊C4—2124no—
339C22⋊C4—2124no—
340C22⋊C4—2121no—
341C22⋊C4—2121no—
342C42—41no—
343C42—41no—
344C42#1171994511yescharacteristic
345C42#1171994511yescharacteristic
346C42D4×D5311yescharacteristic
347C42#1171994911yescharacteristic
348C23—1062no—
349C23—2121no—
350C23—2121no—
351C23—2121no—
352C23—2121no—
353C23—2121no—
354C23—2121no—
355C23#1313318711yescharacteristic
356Q8—4241no—
357Q8—4241no—
358Q8—4241no—
359Q8—4241no—
360D4—1062no—
361D4—1062no—
362D4—2121no—
363D4—2121no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—4241no—
373C2×C4—4241no—
374C2×C4—4241no—
375C2×C4—4241no—
376C2×C4—4241no—
377C2×C4—4241no—
378C2×C4—4241no—
379C2×C4—2122no—
380C2×C4—2121no—
381C2×C4—2121no—
382C2×C4—2122no—
383C2×C4—2121no—
384C2×C4—2122no—
385C2×C4#1313319011yescharacteristic
386C2×C4#1313319011yescharacteristic
387C2×C4#1313319211yescharacteristic
388C2×C4#1313320211yescharacteristic
389C2×C4#1313320211yescharacteristic
390C2×C4#1313320511yescharacteristic
391C2×C4D8×D5311yescharacteristic
392C2×C4#1313323811yescharacteristic
393C2×C4#1313323911yescharacteristic
394C2×C4#1313323911yescharacteristic
395C2×C4#1313324411yescharacteristic
396C2×C4#1313324511yescharacteristic
397C2×C4#1313324911yescharacteristic
398C2×C4#1313325011yescharacteristic
399C8—42no—
400C8—81no—
401C8—2122no—
402C22—4241no—
403C22—4241no—
404C22—4241no—
405C22—4241no—
406C22—2122no—
407C22—2122no—
408C22—2122no—
409C22#1681054111yescharacteristic
410C22#1681075811yescharacteristic
411C22#1681077011yescharacteristic
412C22#1681078711yescharacteristic
413C22#1681078811yescharacteristic
414C22#1681079711yescharacteristic
415C22#1681080711yescharacteristic
416C4—21no—
417C4—21no—
418C4—81no—
419C4—81no—
420C4—8481no—
421C4—4242no—
422C4—4241no—
423C4#1681083911yescharacteristic
424C4#1681084611yescharacteristic
425C4#1681085811yescharacteristic
426C4#1681087211yescharacteristic
427C4#1681093811yescharacteristic
428C4#1681094911yescharacteristic
429C4#1681097811yescharacteristic
430C4#1681098311yescharacteristic
431C2—4242no—
432C2#2027380711yescharacteristic
433C2#2027389011yescharacteristic
434C2#2027395311yescharacteristic
435C2#2027398711yescharacteristic
436C2#2027590011yescharacteristic
437C2#2027593511yescharacteristic
438C2#2027595711yescharacteristic
subgroupG/H
centerC23#13133187
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#1279C2
socleC23×C53#120
prime radicalC22×D212D4
hypercenter#1279D53
hypocenterC53#10416
LCSG ▷ C2×C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1279

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #20275676C2semidirect
3. #20275473C2semidirect
5. C2×#16810370C2semidirect
7. C53×#1279C2semidirect
218. #1279D53semidirect
246. C53#10416semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#202797102
#202756762
#202754732
#202747162
C2×#168103702
C2×#168103522
C53×#12792
#1041653